jmc

To find the decimal expression, we try to get a denominator of the form $2^a\cdot 5^a=10^a$, where $a$ is an integer. $$\frac{3^6}{6^4\cdot 625}=\frac{3^6}{2^4\cdot 3^4\cdot 5^4}=\frac{3^2}{10^4}=9\cdot 10^{-4}=0.0009$$So there are $\boxed{4}$ digits to the right of the decimal point.